Space Coordinates and Vectors in Space. Coordinates in Space

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1 0_110.qd 11//0 : PM Page 77 SECTION 11. Space Coordinates and Vectors in Space 77 -plane Section 11. -plane -plane The three-dimensional coordinate sstem Figure 11.1 Space Coordinates and Vectors in Space Understand the three-dimensional rectangular coordinate sstem. Anale ectors in space. Use three-dimensional ectors to sole real-life problems. Coordinates in Space Up to this point in the tet, ou hae been primaril concerned with the two-dimensional coordinate sstem. Much of the remaining part of our stud of calculus will inole the three-dimensional coordinate sstem. Before etending the concept of a ector to three dimensions, ou must be able to identif points in the three-dimensional coordinate sstem. You can construct this sstem b passing a -ais perpendicular to both the - and -aes at the origin. Figure 11.1 shows the positie portion of each coordinate ais. Taken as pairs, the aes determine three coordinate planes: the -plane, the -plane, and the -plane. These three coordinate planes separate three-space into eight octants. The first octant is the one for which all three coordinates are positie. In this threedimensional sstem, a point P in space is determined b an ordered triple,, where,, and are as follows. directed distance from -plane to P directed distance from -plane to P directed distance from -plane to P Seeral points are shown in Figure (,, ) 1 (,, ) (,, ) (1,, 0) Points in the three-dimensional coordinate sstem are represented b ordered triples. Figure Right-handed sstem Figure 11.1 Left-handed sstem A three-dimensional coordinate sstem can hae either a left-handed or a righthanded orientation. To determine the orientation of a sstem, imagine that ou are standing at the origin, with our arms pointing in the direction of the positie - and -aes, and with the -ais pointing up, as shown in Figure The sstem is right-handed or left-handed depending on which hand points along the -ais. In this tet, ou will work eclusiel with the right-handed sstem. NOTE The three-dimensional rotatable graphs that are aailable in the HM mathspace CD-ROM and the online Eduspace sstem for this tet will help ou isualie points or objects in a three-dimensional coordinate sstem.

2 0_110.qd 11//0 : PM Page CHAPTER 11 Vectors and the Geometr of Space P ( 1, 1, 1 ) d (,, ) Q 1 (,, 1 ) Man of the formulas established for the two-dimensional coordinate sstem can be etended to three dimensions. For eample, to find the distance between two points in space, ou can use the Pthagorean Theorem twice, as shown in Figure B doing this, ou will obtain the formula for the distance between the points 1, 1, 1 and,,. d Distance Formula ( 1 ) + ( 1 ) The distance between two points in space Figure EXAMPLE 1 Finding the Distance Between Two Points in Space The distance between the points, 1, and 1, 0, is d Distance Formula A sphere with center at 0, 0, 0 and radius r is defined to be the set of all points,, such that the distance between,, and 0, 0, 0 is r. You can use the Distance Formula to find the standard equation of a sphere of radius r, centered at 0, 0, 0. If,, is an arbitrar point on the sphere, the equation of the sphere is r (,, ) r Equation of sphere ( 0, 0, 0 ) as shown in Figure Moreoer, the midpoint of the line segment joining the points 1, 1, 1 and,, has coordinates Figure , 1, 1. Midpoint Rule EXAMPLE Finding the Equation of a Sphere Find the standard equation of the sphere that has the points,, and 0,, as endpoints of a diameter. Solution 0 B the Midpoint Rule, the center of the sphere is,,, 1, 0. B the Distance Formula, the radius is r Therefore, the standard equation of the sphere is Midpoint Rule 97. Equation of sphere

3 0_110.qd 11//0 : PM Page 77 SECTION 11. Space Coordinates and Vectors in Space 77 1, 0, 0 0, 0, 1 i k 1,, j 0, 1, 0 The standard unit ectors in space Figure P(p 1, p, p ) Q(q 1, q, q ) = q 1 p 1, q p, q p Figure 11.0 Vectors in Space In space, ectors are denoted b ordered triples 1,,. The ero ector is denoted b 0 0, 0, 0. Using the unit ectors i 1, 0, 0, j 0, 1, 0, and k 0, 0, 1 in the direction of the positie -ais, the standard unit ector notation for is 1 i j k as shown in Figure If is represented b the directed line segment from P p 1, p, p to Q q 1, q, q, as shown in Figure 11.0, the component form of is gien b subtracting the coordinates of the initial point from the coordinates of the terminal point, as follows. 1,, q 1 p 1, q p, q p Vectors in Space Let u u 1, u, u and 1,, be ectors in space and let c be a scalar. 1. Equalit of Vectors: u if and onl if u 1 1, u, and u.. Component Form: If is represented b the directed line segment from P p 1, p, p to Q q 1, q, q, then 1,, q 1 p 1, q p, q p.. Length: 1. Unit Vector in the Direction of : 0 1 1,,,. Vector Addition: u 1 u 1, u, u. Scalar Multiplication: c c 1, c, c NOTE The properties of ector addition and scalar multiplication gien in Theorem 11.1 are also alid for ectors in space. EXAMPLE Finding the Component Form of a Vector in Space Find the component form and magnitude of the ector haing initial point,, 1 and terminal point 0,,. Then find a unit ector in the direction of. Solution The component form of is q 1 p 1, q p, q p 0,, 1 which implies that its magnitude is 7. The unit ector in the direction of is u 1, 7,., 7,

4 0_110.qd 11//0 : PM Page CHAPTER 11 Vectors and the Geometr of Space Recall from the definition of scalar multiplication that positie scalar multiples of a nonero ector hae the same direction as, whereas negatie multiples hae the direction opposite of. In general, two nonero ectors u and are parallel if there is some scalar c such that u c. u u = w = Definition of Parallel Vectors w Two nonero ectors u and are parallel if there is some scalar c such that u c. Parallel ectors Figure 11.1 For eample, in Figure 11.1, the ectors u,, and w are parallel because u and w. EXAMPLE Parallel Vectors Vector w has initial point, 1, and terminal point, 7,. Which of the following ectors is parallel to w? a. u,, 1 b. 1, 1, Solution Begin b writing w in component form. w, 7 1,, 8, a. Because u,, 1 1, 8, 1 w, ou can conclude that u is parallel to w. b. In this case, ou want to find a scalar c such that 1, 1, c, 8,. 1 c c 1 8c c c c Because there is no c for which the equation has a solution, the ectors are not parallel. (1,, ) (, 1, 0) 8 P Q 8 (, 7, ) R The points P, Q, and R lie on the same line. Figure 11. EXAMPLE Using Vectors to Determine Collinear Points Determine whether the points P 1,,, Q, 1, 0, and R, 7, are collinear. Solution The component forms of PQ \ and PR \ are PQ \ 1, 1, 0 1,, and PR \ 1, 7,, 9, 9. These two ectors hae a common initial point. So, P, Q, and R lie on the same line if and onl if PQ \ and PR \ are parallel which the are because PR \ PQ \, as shown in Figure 11..

5 0_110.qd 11//0 : PM Page 777 SECTION 11. Space Coordinates and Vectors in Space 777 EXAMPLE Standard Unit Vector Notation a. Write the ector i k in component form. b. Find the terminal point of the ector 7i j k, gien that the initial point is P,,. Solution a. Because j is missing, its component is 0 and i k, 0,. b. You need to find Q q such that PQ \ 1, q, q 7i j k. This implies that q 1 7, q 1, and q. The solution of these three equations is q 1, q, and q 8. Therefore, Q is,, 8. Application EXAMPLE 7 Measuring Force P (0, 0, ) Q 1, 0 Q 1 (0, 1, 0) Q 1 (,, 0 ) Figure 11. (, ) A teleision camera weighing 10 pounds is supported b a tripod, as shown in Figure 11.. Represent the force eerted on each leg of the tripod as a ector. Solution Let the ectors F 1, F, and F represent the forces eerted on the three legs. From Figure 11., ou can determine the directions of F 1, F, and F to be as follows. PQ \ 1 0 0, 1 0, 0 0, 1, PQ \ 0, 1 0, 0, 1, PQ \ 0, 1 0, 0, 1, Because each leg has the same length, and the total force is distributed equall among the three legs, ou know that F 1 F F. So, there eists a constant c such that F 1 c 0, 1,, F c, 1,, Let the total force eerted b the object be gien b F 10k. Then, using the fact that and F c, 1,. F F 1 F F ou can conclude that F 1, F, and F all hae a ertical component of 0. This implies that c 0 and c 10. Therefore, the forces eerted on the legs can be represented b F 1 0, 10, 0 F,, 0 F,, 0.

6 0_110.qd 11//0 : PM Page CHAPTER 11 Vectors and the Geometr of Space Eercises for Section 11. In Eercises 1, plot the points on the same three-dimensional coordinate sstem. 1. (a), 1, (b) 1,, 1. (a),, (b),,. (a),, (b),,. (a) 0,, (b), 0, In Eercises and, approimate the coordinates of the points... B In Eercises 7 10, find the coordinates of the point. 7. The point is located three units behind the -plane, four units to the right of the -plane, and fie units aboe the -plane. 8. The point is located seen units in front of the -plane, two units to the left of the -plane, and one unit below the -plane. 9. The point is located on the -ais, 10 units in front of the -plane. 10. The point is located in the -plane, three units to the right of the -plane, and two units aboe the -plane. 7. Center: 0,, 8. Center:, 1, 1 Radius: Radius: 9. Endpoints of a diameter:, 0, 0, 0,, 0 0. Center:,,, tangent to the -plane 11. Think About It What is the -coordinate of an point in the -plane? 1. Think About It What is the -coordinate of an point in the -plane? In Eercises 1, determine the location of a point that satisfies the condition(s) < < > 1. > 0,. < 0,. < 0. > 0 In Eercises 8, find the distance between the points.. 0, 0, 0,,,.,,, 7. 1,,,,,,, 8.,,,,, A A B 1,, See for worked-out solutions to odd-numbered eercises. In Eercises 9, find the lengths of the sides of the triangle with the indicated ertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. 9. 0, 0, 0,,, 1,,, 0.,,, 7, 1,,,, 1. 1,,,, 1,, 1, 1,., 0, 0, 0,, 0, 0, 0,. Think About It The triangle in Eercise 9 is translated fie units upward along the -ais. Determine the coordinates of the translated triangle.. Think About It The triangle in Eercise 0 is translated three units to the right along the -ais. Determine the coordinates of the translated triangle. In Eercises and, find the coordinates of the midpoint of the line segment joining the points.., 9, 7,,,., 0,, 8, 8, 0 In Eercises 7 0, find the standard equation of the sphere. In Eercises 1, complete the square to write the equation of the sphere in standard form. Find the center and radius In Eercises 8, describe the solid satisfing the condition... > < 8 1 > 8 1 In Eercises 9, (a) find the component form of the ector and (b) sketch the ector with its initial point at the origin (,, 1) (,, ) (, 0, ) (0,, 1)

7 0_110.qd 11//0 : PM Page 779 SECTION 11. Space Coordinates and Vectors in Space In Eercises, find the component form and magnitude of the ector u with the gien initial and terminal points. Then find a unit ector in the direction of u. Initial Point.,, 0.,,.,, 1. 1,, (0,, ) (,, 0) Terminal Point, 1, 1, 7,,, 0,, In Eercises 7 and 8, the initial and terminal points of a ector are gien. (a) Sketch the directed line segment, (b) find the component form of the ector, and (c) sketch the ector with its initial point at the origin. 7. Initial point: 1,, 8. Initial point:, 1, Terminal point:,, Terminal point:,, 7 In Eercises 9 and 0, the ector and its initial point are gien. Find the terminal point. 9.,, 0. Initial point: 0,, Initial point: 0,, 1,, 1 In Eercises 1 and, find each scalar multiple of and sketch its graph. 1. 1,,.,, 1 (a) (b) (a) (b) 1 (c) (d) 0 (c) (d) In Eercises 8, find the ector, gien that u 1,,,,, 1, and w, 0,.. u. u w. u w. u 1 w 7. u w 8. u w 0 In Eercises 9 7, determine which of the ectors is (are) parallel to. Use a graphing utilit to confirm our results. 9.,, i j k (a),, 10 (a) i j 9k (b),, 10 (b) i j k (c),, 10 (c) 1i 9k (d) 1,, (d) i j 9 8 k (,, ) (,, 0) 71. has initial point 1, 1, and terminal point,,. (a) i 8j k (b) j k 7. has initial point,, 1 and terminal point,,. (a) 7,, (b) 1, 1, In Eercises 7 7, use ectors to determine whether the points are collinear. 7. 0,,,,,,,, 1 7.,, 7,, 0,, 7,, ,,,,, 0, 0, 1, 7. 0, 0, 0, 1,,,,, In Eercises 77 and 78, use ectors to show that the points form the ertices of a parallelogram. 77., 9, 1,, 11,, 0, 10,, 1, 1, 78. 1, 1,, 9, 1,, 11,, 9,,, In Eercises 79 8, find the magnitude of , 0, , 0, 81. i j k 8. i j 7k 8. Initial point of : 1,, Terminal point of : 1, 0, 1 8. Initial point of : 0, 1, 0 Terminal point of : 1,, In Eercises 8 88, find a unit ector (a) in the direction of u and (b) in the direction opposite of u. 8. u, 1, 8. u, 0, u,, 88. u 8, 0, Programming You are gien the component forms of the ectors u and. Write a program for a graphing utilit in which the output is (a) the component form of u, (b) u, (c) u, and (d). 90. Programming Run the program ou wrote in Eercise 89 for the ectors u 1,, and,.,. In Eercises 91 and 9, determine the alues of c that satisf the equation. Let u i j k and i j k. 91. c 9. cu In Eercises 9 9, find the ector with the gien magnitude and direction u. Magnitude Direction u 0,, u 1, 1, 1 u,, 1 9. u,,

8 0_110.qd 11//0 : PM Page CHAPTER 11 Vectors and the Geometr of Space In Eercises 97 and 98, sketch the ector and write its component form. (c) Use a graphing utilit to graph the function in part (a). Determine the asmptotes of the graph. (d) Confirm the asmptotes of the graph in part (c) analticall. (e) Determine the minimum length of each cable if a cable is designed to carr a maimum load of 10 pounds Think About It Suppose the length of each cable in Eercise 109 has a fied length L a, and the radius of each disc is r inches. Make a conjecture about the limit lim T 0 and gie a r 0 a reason for our answer Diagonal of a Cube Find the component form of the unit ector in the direction of the diagonal of the cube shown in the figure. 97. lies in the -plane, has magnitude, and makes an angle of 0 with the positie -ais. 98. lies in the -plane, has magnitude, and makes an angle of with the positie -ais. In Eercises 99 and 100, use ectors to find the point that lies two-thirds of the wa from P to Q. 99. P,, 0, Q 1,, 100. P 1,,, 101. Let u i j, j k, and w au b. (a) Sketch u and. (b) If w 0, show that a and b must both be ero. (c) Find a and b such that w i j k. (d) Show that no choice of a and b ields w i j k. 10. Writing The initial and terminal points of the ector are 1, 1, 1 and,,. Describe the set of all points,, such that. Writing About Concepts Q, 8, 10. A point in the three-dimensional coordinate sstem has coordinates 0, 0, 0. Describe what each coordinate measures. 10. Gie the formula for the distance between the points 1, 1, 1 and,,. 10. Gie the standard equation of a sphere of radius r, centered at 0, 0, State the definition of parallel ectors Let A, B, and C be ertices of a triangle. Find AB \ BC \ CA \ Let r,, and r 0 1, 1, 1. Describe the set of all points,, such that r r Numerical, Graphical, and Analtic Analsis The lights in an auditorium are -pound discs of radius 18 inches. Each disc is supported b three equall spaced cables that are L inches long (see figure). Figure for 111 Figure for Tower Gu Wire The gu wire to a 100-foot tower has a tension of 0 pounds. Using the distances shown in the figure, write the component form of the ector F representing the tension in the wire. 11. Load Supports Find the tension in each of the supporting cables in the figure if the weight of the crate is 00 newtons. cm cm D L T = 1 A C 70 cm B 0 cm 11 cm D ft ft B 100 C A 8 ft 10 ft L 18 in. (a) Write the tension T in each cable as a function of L. Determine the domain of the function. (b) Use a graphing utilit and the function in part (a) to complete the table. Figure for 11 Figure for Construction A precast concrete wall is temporaril kept in its ertical position b ropes (see figure). Find the total force eerted on the pin at position A. The tensions in AB and AC are 0 pounds and 0 pounds. 11. Write an equation whose graph consists of the set of points P,, that are twice as far from A 0, 1, 1 as from B 1,, 0.